Topics in low dimensional topology and Topological Quantum Field Theories

(part of the MRI Masterclass Quantum groups, affine Lie algebras and their applications)

Tought by Michael Müger and Hessel Posthuma.

Location and time:

REC-C 2.10, Mondays 14:00 - 17:00

Very incomplete LECTURE NOTES


Algebra: Linear algebra, Tensor product of vector spaces. Basics of categories. [Lang]
General Topology: Hausdorff, 2nd countability, compactness, connectedness, classification of surfaces (d=2 topological manifolds) [Mun]
Differential Topology: smooth manifolds and maps, tangent space, boundary, orientation, some Morse theory would be helpful [Hir]

Subjects treated

10.9.: Introduction, Motivation, Background. Pedestrian definition of a TQFT [At, L, K]
17.9.: Pedestrian definition of a TQFT (conclusion), example: TQFT in 0+1 dimensions
24.9.: The category of cobordisms. General stuff on tensor categories and functors.
1.10.: More category stuff: monoidal natural transformations, (monoidal) equivalence, symmetries and braidings. TQFT as tensor functor, relation to Atiyah's defin. Group cohomology, in particular H^3(G,A), classification of the group categories C(G,α)
8.10.: (Hessel) TQFTs in 1+1 dimensions vs. commutative Frobenius algebras.
15.10.: TQFTs in d=1+1 (conclusion). Simplicial stuff: semisimplicial sets and their (co)homology, group (co)homology, singular (co)homology.
24.10.: Completion of simplicial stuff.
Homework: All exercises in the lecture notes. (Until november 5.)
29.10.: Final comments on simplicial stuff. Pachner moves. Invariants of 2-manifolds via triangulation [La].
Homework (hand in on 12.11.)
5.11.: TQFT in 1+1 dimensions from triangulation [La,FHK]. Graphical notation for (braided) tensor categories.
Homework (hand in on 19.11.): Exercises 7.16, 7.18, 7.19.
12.11.: The commutative Frobenius algebra of a triangulation TQFT (d=1+1). Pacher moves in d>=3.
Homework (hand in on 26.11.): Exercises 6.30, 6.39
19.11.: Final comments on triangulation in d=1+1. More on categories: semisimplicity, duals, sphericity.
Homework (hand in on 26.11.): Exercises 6.25 and 6.51.
26.11.: 3-Manifold invariant from triangulation: Definition and triangulation-independence (overview). Comments on the triangulation TQFT. [BW,GK]
Homework (hand in on 3.12.): The two exersices in Section 7.2 and Exercises A.65 and A.66 (if you haven't done them already).
3.12. (Hessel): TQFTs in all dimensions I [FQ]. LECTURE NOTES
Homework (hand in on 10.12.): All (four) exercises in the notes.
10.12. (Hessel): TQFTs in all dimensions II [FQ]
17.12. (Hessel/Michael): Outlook: TQFT in 2+1 dimensions from surgery (Reshetikhin/Turaev) [T,BK]


Brief lecture notes on singular homology


[Lang] S. Lang: Algebra. 3rd ed. Addison Wesley or Springer
General topology:
[Mun] J.R. Munkres: Topology. 2nd ed., Prentice Hall, 2000.
Algebraic topology:
J.P. May: A concise course in algebraic topology. Chicago University Press, 1999.
A. Hatcher: Algebraic topology. Cambridge University Press, 2002.
G.E. Bredon: Topology and geometry. Springer, 1993.
Differential topology:
[Hir] M.W. Hirsch: Differential topology. Springer.

Low dimensional topology and TQFT:
[K]: J. Kock: Frobenius algebras and 2D topological quantum field theories. Cambridge University Press.
[PS] V.V. Prasolov and A.B. Sossinsky: Knots, links, braids and 3-manifolds. American Mathematical Society.
[T] V. Turaev: Quantum invariants of knots and 3-manifolds. Walter de Gruyter.
[BK] B. Bakalov and A. Kirillov Jr.: Lectures on tensor categories and modular functors. American Mathematical Society
[Kas] C. Kassel: Quantum groups. Springer.
[KRT] C. Kassel, M. Rosso and V. Turaev: Quantum groups and knot invariants. SMF.
[Li] W.B.R. Lickorish: An introduction to knot theory. Springer.


Two useful reviews by C. Blanchet and V. Turaev: here and here.
[A] M. Atiyah: Topological Quantum Field Theory. Publ. Math. IHES 68, 175-186 (1988).
[BW1] J. W. Barrett and B. W. Westbury: Spherical categories. Adv. Math. 143, 357-375 (1999)
[BW2] J. W. Barrett and B. W. Westbury: Invariants of piecewise-linear 3-manifolds. Trans. Amer. Math. Soc. 348, 3997-4022 (1996)
[FQ] D.S. Freed and F. Quinn: Chern-Simons theory with finite gauge group. Commun. Math. Phys. 156, 435-472 (1993)
[FHK] M. Fukuma, S. Hosono and H. Kawai: Lattice topological field theory in two dimensions. Commun. Math. Phys. 161, 157-175 (1994)
[GK] S. Gelfand and D. Kazhdan: Invariants of three-dimensional manifolds. Geom. Funct. Anal. 6, 268-300 (1996).
[La] R. Lawrence: An introduction to Topological Field Theory. Proc. Symp. Pure Math 51, 89-128 (1996).
[Q] F. Quinn: Lectures on axiomatic topological quantum field theory. In: Daniel S. Freed and Karen K. Uhlenbeck (eds.): Geometry and quantum field theory. IAS/Park City Mathematics Series, 1. American Mathematical Society, Providence, RI, 1995.